# How to show that $u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))$?

If $u(t,x)\in L^{2}([0,T],H^{2}(\mathbb{R}^{n}))$, $\partial_{t}u \in L^{2}([0,T],L^{2}(\mathbb{R}^{n}))$, prove that $$u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))$$

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Do you have a PDE that $u$ satisfies? Evans' PDE book has results of this kind. –  fouryear Jan 17 '13 at 11:19
just to be precise, you find the theorem you are looking for on page 288 (Evans, Partial Differential Equation), theorem 4 for a general non negative integer $m$. –  user8 Jan 17 '13 at 11:29